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Theorem dsmm0cl 18638
Description: The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
Hypotheses
Ref Expression
dsmmcl.p  |-  P  =  ( S X_s R )
dsmmcl.h  |-  H  =  ( Base `  ( S  (+)m  R ) )
dsmmcl.i  |-  ( ph  ->  I  e.  W )
dsmmcl.s  |-  ( ph  ->  S  e.  V )
dsmmcl.r  |-  ( ph  ->  R : I --> Mnd )
dsmm0cl.z  |-  .0.  =  ( 0g `  P )
Assertion
Ref Expression
dsmm0cl  |-  ( ph  ->  .0.  e.  H )

Proof of Theorem dsmm0cl
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dsmmcl.p . . . 4  |-  P  =  ( S X_s R )
2 dsmmcl.i . . . 4  |-  ( ph  ->  I  e.  W )
3 dsmmcl.s . . . 4  |-  ( ph  ->  S  e.  V )
4 dsmmcl.r . . . 4  |-  ( ph  ->  R : I --> Mnd )
51, 2, 3, 4prdsmndd 15822 . . 3  |-  ( ph  ->  P  e.  Mnd )
6 eqid 2441 . . . 4  |-  ( Base `  P )  =  (
Base `  P )
7 dsmm0cl.z . . . 4  |-  .0.  =  ( 0g `  P )
86, 7mndidcl 15807 . . 3  |-  ( P  e.  Mnd  ->  .0.  e.  ( Base `  P
) )
95, 8syl 16 . 2  |-  ( ph  ->  .0.  e.  ( Base `  P ) )
101, 2, 3, 4prds0g 15823 . . . . . . . . . 10  |-  ( ph  ->  ( 0g  o.  R
)  =  ( 0g
`  P ) )
1110, 7syl6eqr 2500 . . . . . . . . 9  |-  ( ph  ->  ( 0g  o.  R
)  =  .0.  )
1211adantr 465 . . . . . . . 8  |-  ( (
ph  /\  a  e.  I )  ->  ( 0g  o.  R )  =  .0.  )
1312fveq1d 5854 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  (  .0.  `  a
) )
14 ffn 5717 . . . . . . . . 9  |-  ( R : I --> Mnd  ->  R  Fn  I )
154, 14syl 16 . . . . . . . 8  |-  ( ph  ->  R  Fn  I )
16 fvco2 5929 . . . . . . . 8  |-  ( ( R  Fn  I  /\  a  e.  I )  ->  ( ( 0g  o.  R ) `  a
)  =  ( 0g
`  ( R `  a ) ) )
1715, 16sylan 471 . . . . . . 7  |-  ( (
ph  /\  a  e.  I )  ->  (
( 0g  o.  R
) `  a )  =  ( 0g `  ( R `  a ) ) )
1813, 17eqtr3d 2484 . . . . . 6  |-  ( (
ph  /\  a  e.  I )  ->  (  .0.  `  a )  =  ( 0g `  ( R `  a )
) )
19 nne 2642 . . . . . 6  |-  ( -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) )  <->  (  .0.  `  a )  =  ( 0g `  ( R `
 a ) ) )
2018, 19sylibr 212 . . . . 5  |-  ( (
ph  /\  a  e.  I )  ->  -.  (  .0.  `  a )  =/=  ( 0g `  ( R `  a )
) )
2120ralrimiva 2855 . . . 4  |-  ( ph  ->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
22 rabeq0 3789 . . . 4  |-  ( { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/)  <->  A. a  e.  I  -.  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) )
2321, 22sylibr 212 . . 3  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  =  (/) )
24 0fin 7745 . . 3  |-  (/)  e.  Fin
2523, 24syl6eqel 2537 . 2  |-  ( ph  ->  { a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin )
26 eqid 2441 . . 3  |-  ( S 
(+)m  R )  =  ( S  (+)m  R )
27 dsmmcl.h . . 3  |-  H  =  ( Base `  ( S  (+)m  R ) )
281, 26, 6, 27, 2, 15dsmmelbas 18637 . 2  |-  ( ph  ->  (  .0.  e.  H  <->  (  .0.  e.  ( Base `  P )  /\  {
a  e.  I  |  (  .0.  `  a
)  =/=  ( 0g
`  ( R `  a ) ) }  e.  Fin ) ) )
299, 25, 28mpbir2and 920 1  |-  ( ph  ->  .0.  e.  H )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   {crab 2795   (/)c0 3767    o. ccom 4989    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Fincfn 7514   Basecbs 14504   0gc0g 14709   X_scprds 14715   Mndcmnd 15788    (+)m cdsmm 18629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-hom 14593  df-cco 14594  df-0g 14711  df-prds 14717  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-dsmm 18630
This theorem is referenced by:  dsmmsubg  18641
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